EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 3, 2023, 1647-1662 ISSN 1307-5543 – ejpam.com Published by New York Business Global 2-Locating Sets in a Graph Gymaima Cañete1, Helen Rara2, Angelica Mae Mahistrado1,∗ 1 Department of Mathematics and Statistics, College of Science and Mathematics, Mindanao State University-Iligan Institute of Technology, 9200 Iligan City, Philippines 2 Department of Mathematics and Statistics, College of Science and Mathematics, Center of Graph Theory, Algebra, and Analysis-Premier Research Institute of Science and Mathematics, Mindanao State University-Iligan Institute of Technology, 9200 Iligan City, Philippines Abstract. Let G be an undirected graph with vertex-set V (G) and edge-set E(G), respectively. A set S ⊆ V (G) is a 2-locating set of G if ∣∣[(NG(x)\NG(y) ) ∩ S]∪ [ ( NG(y)\NG(x) ) ∩ S] ∣∣ ≥ 2, for all x, y ∈ V (G)\S with x ̸= y, and for all v ∈ S and w ∈ V (G)\S, ( NG(v)\NG(w) ) ∩ S ̸= ∅ or( NG(w)\NG[v] ) ∩ S ̸= ∅. In this paper, we investigate the concept and study 2-locating sets in graphs resulting from some binary operations. Specifically, we characterize the 2-locating sets in the join, corona, edge corona and lexicographic product of graphs, and determine bounds or exact values of the 2-locating number of each of these graphs. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: 2-locating set, 2-locating number, join, corona, edge corona, lexico- graphic product 1. Introduction Resolving sets and metric basis are emphasized for their application in computer sci- ence, medical sciences and chemistry. The locating set in graphs can be viewed as the set of monitors that can determine the exact location of an intruder. The concept of 2-locating set is obtained from the concept of locating set. Requiring such a set to be 2-locating implies that every pair of vertices where there is no monitor must be connected to at least two monitoring devices that are connected to other monitors. Also, for every vertex and monitoring device there exists at least one monitor that is connected to it. Hence, 2-locating set can be viewed as the set of monitors that can determine the presence of an intruder. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i3.4821 Email addresses: gymaima.canete@g.msuiit.edu.ph (G. Canete), helen.rara@g.msuiit.edu.ph (H. Rara), angelicamae.mahistrado@g.msuiit.edu.ph (A.M. Mahistrado) https://www.ejpam.com 1647 © 2023 EJPAM All rights reserved. G.Cañete, H. Rara, A.M. Mahistrado / Eur. J. Pure Appl. Math, 16 (3) (2023), 1647-1662 1648 In 1975, Slater [19] introduced the concept of locating sets and its minimum cardinality as locating number. Harary and Melter also utilized a similar idea, although they referred to the locating set and the locating number, respectively, using the terms resolving set and metric dimension. Resolving sets and locating sets, however, are defined differently in more recent studies. In 2013, Bailey et al. [1] defined a resolving set as a set of vertices S in a graph G such that for any two vertices u, v, there exists x ∈ S such that the distance d(u, x) ̸= d(v, x). On the other hand, Canoy and Malacas [8] defined a locating set as a set S ⊆ V (G) of G such that for every two distinct vertices u and v of V (G) \ S, NG(u)∩S ̸= NG(v)∩S. Other variations of locating sets are studied in [16], [6], [11], [13], [14], [15] and [7]. In 2021, J. Cabaro and H. Rara [5] studied the idea of the 2-resolving sets in the join and corona of graphs wherein they introduced the idea of 2-locating sets. This work is therefore motivated by the recent studies on these variations of 2-resolving set and 2-metric dimension that utilize the concepts of 2-locating set and 2-locating number. Other studies that deal with the concept of 2-locating sets are located in [6], [9], [10], [12] and [18]. 2. Terminology and Notation In this study, we consider finite, simple, connected, undirected graphs. For basic graph- theoretic concepts, we then refer readers to [3] and [4]. The following concepts are found in [2], [3], [5] and [17] respectively. The open neighborhood of a vertex v in a graph G is defined as the set NG(v) = {u ∈ V (G) : uv ∈ E(G)}, while the closed neighborhood of a vertex v in G is defined as NG[v] = NG(v) ∪ {v}. The open neighborhood of a set S ⊆ V (G) is defined as NG(S) = ⋃ v∈X NG(v), while its closed neighborhood is NG[S] = NG(S)∪S. A connected graph G of order n ≥ 3 is point distinguishing if for any two distinct vertices u and v of G, NG[u] ̸= NG[v]. It is totally point determining if for any two distinct vertices u and v of G, NG(u) ̸= NG(v) and NG[u] ̸= NG[v]. For an ordered set of vertices W = {w1, w2, ..., wk} ⊆ V (G) and a vertex v in G, we refer to the k-vector (ordered k-tuple) rG(v/W ) = (dG(v, w1), dG(v, w2), ..., dG(v, wk)) as the (metric) representation of v with respect to W . The set W is called a resolving set for G if distinct vertices have distinct representations with respect to W . Hence, if W is a resolving set of cardinality k for a graph G of order n, then the set {rG(v/W ) : v ∈ V (G)} consists of n distinct k-vectors. A resolving set of minimum cardinality is called aminimum resolving set or a basis, and the cardinality of a basis for G is the dimension dim(G) of G. An ordered set of vertices W = {w1, ..., wk} is a k-resolving set for G if, for any distinct vertices u, v ∈ V (G), the (metric) representations rG(u/W ) and rG(v/W ) of u and v, respectively, differ in at least k positions. If k = 1, then the k-resolving set is called a resolving set for G. If k = 2, then the k-resolving set is called a 2-resolving set for G. If G has a k-resolving set, the minimum cardinality dimk(G) of a k-resolving set is called G.Cañete, H. Rara, A.M. Mahistrado / Eur. J. Pure Appl. Math, 16 (3) (2023), 1647-1662 1649 the k-metric dimension of G. If G has a 2-resolving set, we denote the least size of a 2-resolving set by dim2(G) is called a 2-metric dimension of G. A resolving set of size dim2(G) is called a 2-metric basis for G. Let G be any nontrivial connected graph and S ⊆ V (G). A set S ⊆ V (G) is a 2-locating set of G if it satisfies the following conditions: (i) ∣∣[(NG(x)\NG(y) ) ∩S]∪ [ ( NG(y)\NG(x) ) ∩S] ∣∣ ≥ 2, for all x, y ∈ V (G)\S with x ̸= y. (ii) ( NG(v)\NG(w) ) ∩ S ̸= ∅ or ( NG(w)\NG[v] ) ∩ S ̸= ∅, for all v ∈ S and for all w ∈ V (G)\S. The 2-locating number of G, denoted by ln2(G), is the smallest cardinality of a 2-locating set of G. A 2-locating set of G of cardinality ln2(G) is referred to as an ln2-set of G. A set S ⊆ V (G) is a (2, 2)-locating ((2, 1)-locating, respectively) set in G if S is 2- locating and |NG(y)∩S| ≤ |S|−2 (|NG(y)∩S| ≤ |S|−1, respectively), for all y ∈ V (G). The (2, 2)-locating ( (2, 1)-locating, respectively) number ofG, denoted by ln(2,2)(G) (ln(2,1)(G), respectively), is the smallest cardinality of a (2, 2)-locating ((2, 1)-locating, respectively) set in G. A (2, 2)-locating ((2, 1)-locating, respectively) set in G of cardinality ln(2,2)(G) (ln(2,1)(G), respectively) is referred to as an ln(2,2)-set (ln(2,1)-set, respectively) in G. 3. Known Results The following known results are taken from [5]. Remark 1. For any connected nontrivial graph G of order n ≥ 2, 2 ≤ ln2(G) ≤ n. Moreover, ln2(Kn) = n, for n ≥ 2. Theorem 1. Let G be a connected nontrivial graph. Then ln2(G) = 2 if and only if G ∼= P2 or G ∼= P3. Remark 2. Let S ⊆ V (G) For any pair of vertices x, y ∈ S, r(x/S) and r(y/S) differ in at least 2 positions. Hence, to prove that S is a 2-resolving set in G, we only need to show that for every pair of vertices x, y ∈ V (G) where x ∈ S and y ∈ V (G) \ S or both x, y ∈ V (G) \ S, r(x/S) and r(y/S) differ in at least 2 positions. Remark 3. Every 2-locating set in G is a 2-resolving set in G. However, a 2-resolving set in G need not be a 2-locating set in G. Thus, dim2(G) ≤ ln2(G). 4. Preliminary Results Every nontrivial connected graph G admits a 2-locating set. Indeed, the vertex-set of G is a 2-locating set. Proposition 1. For any connected graph G of order n ≥ 2, 2 ≤ ln2(G) ≤ n. Moreover, G.Cañete, H. Rara, A.M. Mahistrado / Eur. J. Pure Appl. Math, 16 (3) (2023), 1647-1662 1650 (i) ln2(G) = 2 if and only if G = K2 or G = P3; (ii) if G = Kn , then ln2(G) = n; (iii) if n = 3 , then ln2(G) = 3 if and only if G = K3; and (iv) if n = 4 , then ln2(G) = 4 if and only if G ∈ {C4,K4, T}. Otherwise, ln2(G) = 3 if and only if G ∈ {P4,K1,3, T ′} where T and T ′ are graphs shown in Figure 1. .................................................................................................................................................................. ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ . ........................................................................................................................................................................... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........................................................................................................................................................................................................................................... ........................................................................................................ ........... .......... .......... .......... .......... .......... .......... .......... .......... ............................................................................................... ......... ........ ........ ........ ........ ........ ........ ........ ........ ... .................................... .................................... .................................... .................................... .................................... ........................................................................ .................................... T : T ′ : Figure 1: Graphs T and T ′ Proof. From Theorem 1 and Remark 1, (i) and (ii) hold. From (ii), (iii) holds. (iv) Suppose n = 4, then the possible connected isomorphic graphs are K4, P4, C4, K1,3, T and T ′. Thus, it can be verified that ln2(G) = 4 if and only if G ∈ {K4, C4, T} and ln2(G) = 3 if and only if G ∈ {P4,K3, T ′}. Remark 4. [5] Every 2-locating set of a connected graph G is 2- resolving. Thus, dim2(G) ≤ ln2(G). Theorem 2. Let a and b be any positive integers such that 2 ≤ a ≤ b. Then there exists a connected graph G such that dim2(G) = a and ln2(G) = b. Proof. Suppose that a = b. Consider the graph G = Ka. Then dim2(G) = ln2(G) = a. Next, suppose that a < b. Consider the following cases: Case 1: a = 2 Let m = b − a and consider the graph G in Figure 2. Let S1 = {x1, x2} and S2 = S1 ∪ {v1, v2, . . . , vm}. Then S1 and S2 are dim2 − set and ln2 − set of G, respectively. Hence, dim2(G) = a and ln2(G) = a+m = b. ............................................... ............................................... ............................................... ............................................... ............................................... ............................................... ..................................................................................................................................................................................................................................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... v1 v2 vmx1 x2 G : • • • • • . . . Figure 2: A graph G Case 2: a ≥ 3. Let m = b − a and consider the graph G′ in Figure 3. Let S1 = {y1, y2, . . . , ya} and S2 = S1 ∪ {u1, u2, . . . , um}. Then S1 and S2 are dim2-set and ln2-set of G ′, respectively. Hence, dim2(G ′) = a and ln2(G ′) = a+m = b. G.Cañete, H. Rara, A.M. Mahistrado / Eur. J. Pure Appl. Math, 16 (3) (2023), 1647-1662 1651 ......... ........ ........ ........ ........ ...... ............................................... .... ........ ........ ........ ........ ........ ... ............................................... .... ........ ........ ........ ........ ........ ... ......... ........ ........ ........ ........ ...... ................... .................. ................. ...................................................... .......................................... ......... ........ ........ ........ ........ ...... ............................................................................................................................................ .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... u1 um−1 um−2 um−3 um y1 y2 y3 ya G′ : • • • • • • ••• . . . . . . Figure 3: A graph G′ . Corollary 1. For each positive integer n, there exists a connected graph G such that ln2(G)− dim2(G) = n, that is, ln2 − dim2 can be made arbitrarily large. We now characterize the 2-locating sets in some graphs under some binary operations. 5. Join of Graphs This section presents the characterizations on the 2-locating sets in the join of graphs. Theorem 3. [6] Let G and H be nontrivial connected graphs. A proper subset S of V (G+H) is a 2-resolving set in G+H if and only if SG = V (G)∩S and SH = V (H)∩S are 2-locating sets in G and H, respectively, where SG or SH is (2,2)-locating set or SG and SH are (2,1)-locating sets. Theorem 4. [6] Let G be a connected graph of order greater than 3 and let K1 = ⟨v⟩. Then S ⊆ V (K1 +G) is a 2-resolving set of K1 +G if and only if either v /∈ S and S is a (2,2)-locating set in G or S = {v} ∪ T is (2,1)-locating set in G. Theorem 5. Let G and H be connected graphs. Then S ⊆ V (G+H) is a 2-locating set in G +H if and only if S is a 2-resolving set in G +H where S = SG ∪ SH , SG ⊆ V (G) and SH ⊆ V (H). Proof. Suppose S is a 2-locating set in G + H. Let p, q ∈ V (G + H). Consider p, q ∈ V (G + H) \ S or [p ∈ V (G + H) \ S or q ∈ S]. Since S is a 2-locating set, then rG+H(p/S) and rG+H(q/S) differ in at least 2 positions. By Definition of 2-resolving set and Remark 2, S is a 2-resolving set. For the converse, suppose S is a 2-resolving set in G + H. Let S = SG ∪ SH where SG ⊆ V (G) and SH ⊆ V (H). Let p, q ∈ V (G+H) \ S. Consider the following cases. Case 1 p, q ∈ V (G) \ SG. Since S is a 2-resolving set, rG+H(p/S) and rG+H(q/S) differ in at least 2 positions. By definition of G+H, rG(p/SG) and rG(q/SG) differ in at least 2 positions. Since dG+H(p, u) and dG+H(q, u) is either 0,1 or 2 for each u ∈ V (G+H), there exist at least two vertices x, y ∈ SG such that x, y ∈ NG(p) \NG(q) or x, y ∈ NG(q) \NG(p) or x ∈ NG(p) \NG(q) and y ∈ NG(q) \NG(p). Hence,∣∣[(NG+H(p)\NG+H(q) ) ∩ S] ∪ [ ( NG+H(q)\NG+H(p) ) ∩ S] ∣∣ ≥ 2. (1) Case 2. p, q ∈ V (H) \ SH Proof is similar to Case 1. G.Cañete, H. Rara, A.M. Mahistrado / Eur. J. Pure Appl. Math, 16 (3) (2023), 1647-1662 1652 Case 3. p ∈ V (G) \ SG and q ∈ V (H) \ SH Note that rG+H(p/S) = (2, 2, 2, . . . , 1, 1, . . . , 1) and rG+H(q/S) = (1, 1, 1, . . . , 2, 2, . . . , 2). Then there exist x ∈ SG \ NG(p) and y ∈ SH \ NH(q) or ∃w, r ∈ SG \ NG(p) or w, r ∈ SH \NH(q). Hence, inequality (1) holds. Suppose p ∈ S and q ∈ V (G+H) \ S. Consider the following cases. Case 1 p ∈ SG and q ∈ V (G) \ SG. Since rG+H(p/S) and rG+H(q/S) differ in at least 2 positions, by definition of G + H, rG(p/SG) and rG(q/SG) differ in at least 2 positions, This implies that ( NG(p)\NG(q) ) ∩ SG ̸= ∅ or ( NG(q)\NG(p) ) ∩ SG ̸= ∅. Hence, ( NG+H(p)\NG+H(q) ) ∩ S ̸= ∅ or ( NG+H(q)\NG+H(p) ) ∩ S ̸= ∅. Case 2. p ∈ SG and q ∈ V (H) \ SH Note that rG+H(p/S) = (. . . , 0, . . . , 1, 1, . . . , 1) and rG+H(q/S) = (1, 1, . . . , 0, . . .). Hence, there exist at least one vertex x ∈ SG \ NG(p) or there exist at least one vertex y ∈ SH \NG(q). Thus,( NG+H(p)\NG+H(q) ) ∩ S ̸= ∅ or ( NG+H(q)\NG+H(p) ) ∩ S ̸= ∅. The proof that p ∈ V (G+H) \ S and q ∈ S is similar. Therefore, S is a 2-locating set of G+H. The following corollaries follow immediately from Theorem 5, Theorem 4, and Theorem 3. Corollary 2. Let G be a nontrivial connected graph and K1 = ⟨v⟩. Then S ⊆ V (K1+G) is a 2-locating set in K1 +G if and only if it satisfies the following conditions: (i) v /∈ S and S is (2,2)-locating set of G. (ii) S = {v} ∪ T and T is (2,1)-locating set in G. Corollary 3. Let G be any nontrivial connected graph. Then ln2(K1 +G) = min{ln(2,2)(G), ln(2,1)(G) + 1}. Corollary 4. Let G and H be nontrivial connected graphs. A set S ⊆ V (G+H) is a 2- locating set in G+H if and only if S = SG∪SH where SG = V (G)∩S and SH = V (H)∩S are 2-locating sets of G and H, respectively, where SG or SH is a (2, 2)-locating set or SG and SH are (2, 1)-locating sets. Corollary 5. Let G and H be nontrivial connected graphs. Then ln2(G+H) = min{ln(2,2)(G) + ln2(H), ln2(G) + ln(2,2)(H), ln(2,1)(G) + ln(2,1)(H)}. G.Cañete, H. Rara, A.M. Mahistrado / Eur. J. Pure Appl. Math, 16 (3) (2023), 1647-1662 1653 6. Corona of Graphs This section presents the characterizations on the 2-locating sets in the corona of graphs. Theorem 6. Let G and H be nontrivial connected graphs with ∆(H) ≤ |V (H)| − 3. A set S ⊆ V (G ◦H) is a 2-locating set of G ◦H if and only if S = A ∪ ( ⋃ v∈V (G) Sv ) where A ⊆ V (G) and V (Hv) ∩ S ̸= ∅ for each v ∈ V (G) and the following are satisfied (i) Sv is a 2-locating set of Hv for each v ∈ V (G) and Su or Sv is total 2-dominating for u, v ∈ V (G) \A or otherwise, Su and Sv are total dominating; (ii) for each v ∈ V (G) \ A, Sv is a (2, 2)-locating set of Hv with NG(v) ∩A = ∅ and Sv is (2,1)-locating set, otherwise; and (iii) for each v ∈ A, Sv is a (2, 1)-locating set of Hv if NG(v) ∩A = ∅. Proof. Suppose S ⊆ V (G ◦ H) is a 2-locating set in G ◦ H. Let A = V (G) ∩ S, Sv = S ∩ V (Hv) for all v ∈ V (G). Then S = A ∪ ( ⋃ v∈V (G) Sv ) where A ⊆ V (G) and Sv ⊆ V (Hv). Now, suppose Sv = ∅ for some v ∈ V (G). Let x, y ∈ V (Hv)\Sv. Then∣∣[(NHv(x)\NHv(y) ) ∩ Sv] ∪ [ ( NHv(y)\NHv(x) ) ∩ Sv] ∣∣ = 0, for all x, y ∈ V (Hv)\Sv with x ̸= y, a contradiction to the assumption of S. Thus, Sv ̸= ∅ for all v ∈ V (G ◦H). To prove (i), let x, y ∈ V (Hv) where v ∈ V (G). Then x, y ∈ V (G◦H). Since NHv(x) = NG◦H(x)\{v} and NHv(y) = NG◦H(y)\{v}, and S is a 2-locating set, this implies that Sv is also 2-locating set in Hv. Next, suppose Su or Sv is not a total dominating, say Sv is not a total dominating set for some v ∈ V (G) \A. Let x ∈ V (Hu)\Su and y ∈ V (Hv)\Sv. Since S is a 2-locating set, there exist w, z ∈ (NHv(x)\NHv(y)) ∩ Su implying that Su is a total 2-dominating set. To prove (ii), let v ∈ V (G)\A. Suppose NG(v)∩A = ∅. Since Sv ⊆ NG◦H(v) and S is 2-locating, there exist at least two vertices x, y ∈ Sv \NHv(p) for each p ∈ V (Hv). Thus, Sv is (2, 2)- locating set. On the other hand, if NG(v) ∩ A ̸= ∅, there exists at least one vertex z ∈ Sv \NHv(p). This implies that Sv is (2, 1) -locating. To prove (iii), let v ∈ A and NG(v) ∩A = ∅. Since Sv is a 2-locating set, there exists r ∈ Sv \NHv(p) for every p ∈ V (Hv). Thus, Sv is a (2,1)-locating set in Hv. For the converse, suppose S is a set as described and satisfies the given conditions. Let p, q ∈ V (G ◦ H) with p ̸= q and let u, v ∈ V (G) such that p ∈ V (u + Hu) and q ∈ V (v +Hv). Suppose p, q ∈ V (G ◦H)\S. Consider the following cases: Case 1. u = v Subcase 1.1 p, q ∈ V (Hu) \ Su Since Su is a 2-locating set of Hu, NHu(p) = NG◦H(p) and NHu(q) = NG◦H(q). Then |[(NG◦H(q)\NG◦H(p)) ∩ S] ∪ [(NG◦H(p)\NG◦H(q)) ∩ S]| ≥ 2 G.Cañete, H. Rara, A.M. Mahistrado / Eur. J. Pure Appl. Math, 16 (3) (2023), 1647-1662 1654 and for all r ∈ Su, (NG◦H(r)\NG◦H(q)) ∩ S ̸= ∅. Thus, S is a 2-locating set. Subcase 1.2 p = v and q ∈ V (Hv) \ Sv If NG(v)∩A = ∅, by (ii) Sv is a (2,2)-locating set. Hence, there exist at least two distinct vertices x, y ∈ V (Hv) \NHv(q). Thus, x, y ∈ NG◦H(p) \NG◦H(q). If NG(v) ∩A ̸= ∅, then there exists z ∈ (NG◦H(v) ∩A) \NG◦H(q). Since γ(H) ̸= 1, there exists w ∈ Sv \NHv(q). Hence, w, z ∈ NG◦H(p) \NG◦H(q) ∩ S. Thus, |(NG◦H(p) \NG◦H(q) ∩ S)| ≥ 2. Subcase 1.3 q = v and p ∈ V (Hu) \ Su The proof is similar to the proof of Subcase 1.2. Case 2. u ̸= v Subcase 2.1 p ∈ V (Hu) \ Su and q ∈ V (Hv) \ Sv If u, v ∈ A, then we are done. Suppose u, v /∈ A. Since Su and Sv are total dominating, there exist x ∈ (NHu(p) ∩ Su) \ NHv(q) and y ∈ (NHv(q) ∩ Sv) \NHu(p). Subcase 2.2 p = u and q ∈ V (Hv) \ Sv Since p /∈ A, Su is a total dominating set of Hu. Hence, |Su| ≥ 2. Thus, |(NG◦H(p) \ NG◦H(q)) ∩ S| ≥ 2. Suppose p ∈ S and q ∈ V (G ◦H) \ S. Consider the following cases Case 1 u = v Subcase 1.1 p ∈ Sv and q ∈ V (Hv) \ Sv Since Sv is a 2-locating, then (NG◦H(p)\NG◦H(q)) ∩ S ̸= ∅. Subcase 1.2 u = p and q ∈ V (Hv) \ Sv. Then u ∈ A. If NG(p) ∩ A ̸= ∅, then we are done. Suppose NG(p) ∩ A = ∅. Then by (iii), Sv is a (2,1)-locating. Thus, (NG◦H(p) \NG◦H(q)) ∩ S ̸= ∅. Case 2 u ̸= v Subcase 2.1 p ∈ Su and q ∈ V (Hv) \ Sv If u ∈ A or v ∈ A, then we are done. If u, v /∈ A, then by (i) Su and Sv are total dominat- ing. Hence, there exist x ∈ (NG◦H(p) ∩ S) \NG◦H(q) and y ∈ (NG◦H(q) ∩ S) \NG◦H(p). Subcase 2.2 p = u and q ∈ V (Hv) \ Sv Since Su ̸= ∅, (NG◦H(p) ∩ S) \NG◦H(q) ̸= ∅. Subcase 2.3 p ∈ Su and q = v Similar to the proof of Subcase 2.2. Accordingly, S is a 2-locating set of G ◦H. Corollary 6. Let G of order n and H be nontrivial connected graphs with γ(H) ̸= 1. Then (i) ln2(G ◦H) ≤ γt(G) + n · ln2(H); and (ii) If ln2(H) = ln(2,1)(H) = ln(2,2)(H). Then ln2(G ◦H) = n · ln2(H). Proof. (i.) Let S = V (G ◦H) be a 2-locating set in G ◦H. Let A be a γt-set of G and Sv be an ln2-set of H v. Then S = A ∪ ( ⋃ v∈V (G) Sv ) is a 2-locating set of G ◦H. Thus, G.Cañete, H. Rara, A.M. Mahistrado / Eur. J. Pure Appl. Math, 16 (3) (2023), 1647-1662 1655 ln2(G ◦H) ≤ |S| = |A|+ ∑ v∈V (G) |Sv| = γt(G) + |V (G)|(ln2(H)) = γt(G) + n · ln2(H). (ii.) Let A = ∅ and Sv be an ln2-set of Hv. Then S = A ∪ ( ⋃ v∈V (G) Sv ) is a 2-locating set of G ◦H. Thus, ln2(G ◦H) ≤ |S| = ∑ v∈V (G) |Sv| = |V (G)|ln2(H) = n · ln2(H). Next, let S0 be an ln2-set in G ◦ H. Then by Theorem 6, S0 = A0∪ ( ⋃ v∈V (G) Sv ) where A0 ⊆ V (G) and Sv is a 2-locating set of Hv, for all v ∈ V (G). Thus, ln2(G ◦H) = |S0| = |A0|+ | ⋃ v∈V (G) Sv| ≥ ∑ v∈V (G) |Sv| ≥ n · ln2(H) Thus, equality holds. 7. Edge Corona of Graphs This section presents characterizations on the 2-locating sets in the edge corona of graphs. Theorem 7. Let G and H be nontrivial connected graphs where G ̸= P2 and ∆(H) ≤ |V (H)| − 3. A set C ⊆ V (G ⋄ H) is a 2-locating set of G ⋄ H if and only if C is a 2-resolving set of G ⋄H. Proof. Let C be a 2-locating set of G ⋄ H. By Remark 3, C is a 2-resolving set of G ⋄H. G.Cañete, H. Rara, A.M. Mahistrado / Eur. J. Pure Appl. Math, 16 (3) (2023), 1647-1662 1656 Conversely, suppose C is a 2-resolving set of G ⋄ H. Let a, b ∈ V (Huv)\Suv where a ̸= b or [a ∈ Suv and b /∈ Suv]. Since C is a 2-resolving set in G ⋄ H, rG⋄H(a/C) and rG⋄H(b/C) differ in at least 2 positions. Since NG⋄H(a) = NHuv(a)∪{u, v} and NG⋄H(b) = NHuv(b) ∪ {u, v}, rHuv(a/Suv) and rHuv(b/Suv) must differ in at least 2 positions. By definition of G ⋄ H, there exist at least two vertices say p, q ∈ V (Huv) ∩ Suv such that either p, q ∈ NHuv(a)\NHuv(b) or p, q ∈ NHuv(b)\NHuv(a) or p ∈ NHuv(a)\NHuv(b) and q ∈ NHuv(b)\NHuv(a). Similarly, if a ∈ Suv and b ∈ V (Huv)\Suv, then there exists a vertex s ∈ V (Huv) ∩ Suv such that s ∈ NHuv(a)\NHuv(b) or s ∈ NHuv(b)\NHuv(a). Thus, it follows that Suv is a 2-locating set of Huv. Accordingly, C is a 2-locating set in G ⋄H. Theorem 8. Let G and H be any nontrivial connected graphs where G ̸= P2 and ∆(H) ≤ |V (H)| − 3. A set C ⊆ V (G ⋄H) is a 2-locating set of G ⋄H if and only if C = A ∪  ⋃ uv∈E(G) Suv  where (i) A ⊆ V (G), Suv ⊆ V (Huv) and V (Huv) ∩ C ̸= ∅; (ii) Suv ⊆ V (Huv) is a 2-locating set of Huv for all uv ∈ E(G) or if uv is a pendant edge, then Suv is a (2, 1)-locating set of Huv whenever l(⟨{u, v}⟩) ⊆ A and Suv is a (2, 2)-locating set of Huv otherwise. Proof. Suppose that C ⊆ V (G⋄H) is a 2-locating set in G⋄H. Let A = V (G)∩C and Suv = C ∩ V (Huv) for all uv ∈ E(G). Then C = A∪ ( ⋃ uv∈E(G) Suv ) where A ⊆ V (G) and Suv ⊆ V (Huv). Now, suppose that Suv = ∅ for some uv ∈ E(G). Let x, y ∈ V (Huv)\Suv. Then ∣∣[(NHuv(x)\NHuv(y) ) ∩ Suv] ∪ [ ( NHuv(y)\NHuv(x) ) ∩ Suv] ∣∣ = 0, a contradiction to the assumption of C. Thus, Suv ̸= ∅ for all uv ∈ E(G). Next, we claim that Suv is a 2-locating set in Huv for each uv ∈ E(G). Let a, b ∈ V (Huv) where uv ∈ E(G). Then a, b ∈ V (G ⋄H). Since NHuv(a) = NG⋄H(a) \ {u, v} and NHuv(b) = NG⋄H(b) \ {u, v} and C is a 2-locating set, this implies that Suv is also a 2-locating set in Huv. Next, suppose that uv is a pendant edge and suppose u is an end-vertex. Then ⟨u⟩+Huv is a subgraph of G⋄H. Since Suv = C ∩V (Huv) ⊆ C and C is a 2-locating set, it follows by Corollary 2, Suv is a (2,1)-locating set of Huv whenever u ∈ C and Suv is a (2,2)-locating set of Huv, otherwise. Conversely, let C be the set as described and satisfies the given conditions. Let x, y ∈ V (G⋄H) with x ̸= y. Then it can be easily verified that rG⋄H(x/C) and rG⋄H(y/C) differ in at least two positions for all x, y ∈ V (G) or x ∈ V (Huv) and y ∈ V (G) for all edges uv ∈ E(G) or x ∈ V (Hpq) and y ∈ V (Hab), for some pq, ab ∈ E(G). Hence, consider only the following cases: G.Cañete, H. Rara, A.M. Mahistrado / Eur. J. Pure Appl. Math, 16 (3) (2023), 1647-1662 1657 Case 1: x, y ∈ V (Huv) \ Suv or x ∈ V (Huv) \ Suv and y ∈ Suv for some edge uv ∈ E(G). Now, since Suv is 2-locating set, rHuv(x/Suv) and rHuv(x/Suv) differ in at least two po- sitions. Then by definition of G ⋄ H, rG⋄H(x/C) and rG⋄H(y/C) differ in at least two positions. Case 2: x ∈ V (Huv) \ Suv or x ∈ Suv and y = u for some pendant edge uv ∈ E(G) and u is an endvertex Since Suv is a (2,2)-locating set, there exists a, b ∈ Suv \ NHuv(x) but a, b ∈ NG⋄H(y). Thus, it follows that rG⋄H(x/C) and rG⋄H(y/C) differ at ath and bth positions. Therefore, C is a 2-resolving set in G ⋄H. By Theorem 7, C is a 2-locating set in G. Corollary 7. Let G and H be any nontrivial connected graphs where G ̸= P2 with |E(G)| = m and ∆(H) ≤ |V (H)| − 3. Then the following statements hold. (i) If G is a graph with no pendant edges, then ln2(G ⋄H) = m · ln2(H). (ii) If G is a graph with k ≥ 1 pendant edges, then ln2(G ⋄H) = min {( m− k ) ln2(H) + k · ln(2,1)(H) + k, ( m− k ) ln2(H) +k · ln(2,2)(H) } and ln2(G ⋄ H) = ( m − k ) ln2(H) + k · ln(2,2)(H) whenever ln(2,2)(H) = ln(2,1)(H). Proof. (i) Suppose G is a graph with no pendant edges. Now, set A = ∅ and let Suv be an ln2 − set of Huv for all uv ∈ E(G). Then C = A∪ ( ⋃ uv∈E(G) Suv ) is a 2-locating set in G ⋄H by Theorem 8. Hence, ln2(G ⋄H) ≤ |C| = |A|+ |E(G)||Suv| = m(ln2(H)). Next, let C0 be an ln2 − set in G ⋄ H. Then by Theorem 8, C0 = A0 ∪ ( ⋃ uv∈E(G) Suv ) where A0 ⊆ V (G) and Suv is a 2-locating set of Huv for all uv ∈ E(G). Thus, ln2(G ⋄H) = |C0| = |A0|+ | ⋃ uv∈E(G) Suv| ≥ ∑ uv∈E(G) |Suv| ≥ m · ln2(H). Therefore, ln2(G ⋄H) = m · ln2(H). (ii) Let G be a graph with pendant edges and A ⊆ V (G) consists of pendant edges in a graph G, that is |A| = k. By Theorem 8, Suv is a 2-locating set of Huv for all uv ∈ E(G) and Suv is a (2,1)-locating set of Huv whenever l(uv) ⊆ A and Suv is a (2,2)-locating set G.Cañete, H. Rara, A.M. Mahistrado / Eur. J. Pure Appl. Math, 16 (3) (2023), 1647-1662 1658 of Huv, otherwise. If Suv is a (2,2)-locating sets in Huv, then (m− k)ln2(H) + k · ln(2,2)(H) ≤ |C| = ln2(G ⋄H). If Suv is a (2,2)-locating sets in Huv, then (m− k)ln2(H) + k · ln(2,1)(H) + k ≤ |C| = ln2(G ⋄H). Thus, ln2(G ⋄H) ≥ min{(m− k)ln2(H) + k · ln(2,1)(H) + k, (m− k)ln2(H) + k · ln(2,2)(H)}. Let (m − k)ln2(H) + k · ln(2,1)(H) + k ≤ (m − k)ln2(H) + k · ln(2,2)(H). Let Suv be the minimum (2,1)-locating set in Huv whenever l(uv) ⊆ A and Suv be the min- imum (2,2)-locating set in Huv, otherwise. Then, C is a 2-locating set in G ⋄ H by Corollary 7. Hence, ln2(G ⋄ H) ≤ |C| = (m − k)ln2(H) + k · ln(2,1)(H) + k. Sim- ilarly, if (m − k)ln2(H) + k · ln(2,2)(H) ≤ (m − k)ln2(H) + k · ln(2,1)(H) + k. Then ln2(G ⋄H) ≤ |C| = (m− k)ln2(H) + k · ln(2,2)(H). Thus, ln2(G ⋄H) ≤ min{(m− k)ln2(H) + k · ln(2,1)(H) + k, (m− k)ln2(H) + k · ln(2,2)(H)}. Therefore, ln2(G ⋄H) = min{(m− k)ln2(H) + k · ln(2,1)(H) + k, (m− k)ln2(H) + k · ln(2,2)(H)}. 8. Lexicographic Product of Graphs This section presents characterizations on the 2-locating sets in the lexicographic prod- uct of graphs. Theorem 9. [6] Let G and H be nontrivial connected graphs. Then W = ⋃ x∈S [{x} × Tx], where S ⊆ V (G) and Tx ⊆ V (H) for each x ∈ S, is a 2-resolving set in G[H] if and only if (i) S = V (G); (ii) Tx is a 2-locating set in H for every x ∈ V (G); (iii) Tx and Ty are (2,1)-locating sets or one of Tx and Ty is a(2,2)-locating set in H whenever x, y ∈ EQ1(G); and G.Cañete, H. Rara, A.M. Mahistrado / Eur. J. Pure Appl. Math, 16 (3) (2023), 1647-1662 1659 (iv) Tx and Ty are (2-locating)dominating sets in H or one of Tx and Ty is a 2-dominating set whenever x, y ∈ EQ2(G). Theorem 10. Let G and H be nontrivial connected graphs with ∆(H) ≤ |V (H)| − 3. Then W = ⋃ x∈S [{x}×Tx], where S ⊆ V (G) and Tx ⊆ V (H) for each x ∈ S, is a 2-locating set in G[H] if and only if it is a 2-resolving set and it satisfies the following: (i) S = V (G); (ii) Tx is a 2-locating set in H for every x ∈ V (G); (iii) Tx and Ty is a (2, 1)-locating set or one of Tx and Ty is a (2, 2)-locating set in H whenever x, y ∈ V (G) with NG[x] = NG[y]; and (iv) Tx and Ty are (2 − locating) dominating sets in H or one of Tx and Ty is a 2- dominating set whenever x, y ∈ V (G) with dG(x, y) = 2 and NG(x) = NG(y). Proof. Suppose W = ⋃ x∈S [ {x} × Tx ] is a 2-locating set in G[H]. Suppose there exists x ∈ V (G)\S. Pick a, b ∈ V (H), where a ̸= b. Then (x, a), (x, b) /∈ W and (x, a) ̸= (x, b). Since x /∈ S, (x, r) ∈ V (G[H]) \ W . Note that (z, c) ∈ NG[H](x, a) ∪ NG[H](x, b) for all z ∈ NG(x). Thus, NG[H](x, a) \NG[H](x, b) = ∅ and NG[H](x, b) \NG[H](x, a) = ∅. This implies that W is not a 2-locating set of G[H], a contradiction to the assumption on W . Therefore, S = V (G). To prove (ii), let x ∈ V (G) and p, q ∈ V (H) where p ̸= q. Then (x, p) ̸= (x, q). If p, q /∈ Tx or [p ∈ Tx and q /∈ Tx], then (x, p), (x, q) /∈ W or [(x, p) ∈ W and (x, q) /∈ W ]. Since W is a 2-locating set in G[H], by definition of G[H] there exist at least two ver- tices (x, r), (x, s) ∈ V (H) ∩ Tx such that either (x, r), (x, s) ∈ NH((x, p))\NH((x, q)) or (x, r), (x, s) ∈ NH((x, q))\NH((x, p)) or (x, r) ∈ NH((x, p))\NH((x, q)) and (x, s) ∈ NH((x, q))\NH((x, p)). Similarly, if (x, p) ∈ W and (x, q) /∈ W , then there exists a vertex t ∈ V (H) ∩ Tx such that (x, t) ∈ NH((x, p))\NH((x, q)) or (x, t) ∈ NH((x, q))\NH((x, p)). Therefore, it follows that Tx is a 2-locating set of H for every x ∈ V (G). Thus, (ii) follows. To prove (iii), let x, y ∈ V (G) with NG[x] = NG[y]. Let a, b ∈ V (H), a ̸= b. Since W is a 2-locating set, it is not possible that NH(a) ∩ Tx = Tx and NH(b) ∩ Ty = Ty. If Tx or Ty is (2, 2)-locating, then we are done. Otherwise, Tx and Ty are (2, 1)-locating. To prove (iv), let x, y ∈ V (G) where dG(x, y) = 2 and NG(x) = NG(y). Let a, b ∈ V (H), a ̸= b. Suppose one of Tx and Ty, say Tx is not a dominating set in H. Pick a ∈ V (H)\NH [Tx] and let b ∈ V (H)\Ty. Since dG[H]((x, a), (y, b)) = 2, for all (y, b), it follows that |NH(b) ∩ Ty| ≥ 2, i.e., Ty is a 2-dominating set. Conversely, let W be the set as described and satisfies the given conditions. Let (x, a), (y, b) ∈ V (G[H]), (x, a) ̸= (y, b). Consider the following cases. Case 1. x = y Suppose (x, a), (y, b) /∈ W . Then a ̸= b and a, b /∈ Tx = Ty. By (ii), Tx is a 2-locating set. On the other hand, if (x, a) ∈ W , (y, b) /∈ W , then a ∈ Tx, b /∈ Ty. Since Tx is a 2-locating set, there exists (x, s) ∈ V (H) ∩ Tx such that (x, s) ∈ NH((x, a))\NH((y, b)) or (x, s) ∈ NH((y, b))\NH((x, a)). Thus, it follows that W is a 2-locating set of G[H]. G.Cañete, H. Rara, A.M. Mahistrado / Eur. J. Pure Appl. Math, 16 (3) (2023), 1647-1662 1660 Case 2. x ̸= y. Subcase 2.1 xy ∈ E(G). If NG[x] ̸= NG[y], then we are done. Suppose NG[x] = NG[y], then by (iii), Tx and Ty are (2, 1)-locating sets in H or one of Tx and Ty is a (2, 2)-locating set in H. Subcase 2.2 xy /∈ E(G) If dG(x, y) > 2, then we are done. Suppose dG(x, y) = 2 and NG(x) = NG(y). Suppose (x, a), (y, b) /∈ W . Then a /∈ Tx and y /∈ Ty. If Tx and Ty are both domi- nating, then there exist at least two vertices (x, r), (x, s) ∈ V (H) ∩ Tx such that either (x, r), (x, s) ∈ NH((x, p))\NH((x, q)) or (x, r), (x, s) ∈ NH((x, q))\NH((x, p)) or (x, r) ∈ NH((x, p))\NH((x, q)) and (x, s) ∈ NH((x, q))\NH((x, p)). If one, say Ty, is a 2-dominating set, then there exist at least two vertices r, s ∈ V (H) ∩ Tx such that either (x, r), (x, s) ∈ NH((x, p))\NH((x, q)) or (x, r), (x, s) ∈ NH((x, q))\NH((x, p)) or (x, r) ∈ NH((x, p))\NH((x, q)) and (x, s) ∈ NH((x, q))\NH((x, p)). Similarly, if (x, a) ∈ W , (y, b) /∈ W , there exists (x, s) ∈ V (H)∩Tx such that (x, s) ∈ NH((x, a))\NH((y, b)) or (x, s) ∈ NH((y, b))\NH((x, a)). Accordingly, W is a 2-locating set of G[H]. Corollary 8. Let G and H be nontrivial connected graphs with ∆(H) ≤ |V (H)| − 3. If G is a totally point determining graph, then ln2(G[H]) = |V (G)| · ln2(H). Proof. Supppose that G is totally point determining graph. Let S = V (G) and let Tx be an ln2-set of H for each x ∈ S. By Theorem 10, W = ⋃ x∈S [{x} × Tx] is a 2-locating set of G[H]. It follows that ln2(G[H]) ≤ |W | = |V (G)||Tx| = |V (G)| · ln2(H). Now, if W0 = ⋃ x∈S0 [{x} × Tx] is an ln2-set of G[H], then S0 = V (G) and Tx is a 2-locating set of H for each x ∈ V (G) by Theorem 10. Hence, ln2(G[H]) = |W0| = |V (G)||Tx| ≥ |V (G)| · ln2(H). Therefore, ln2(G[H]) = |V (G)| · ln2(H). 9. Conclusion It is shown that the difference of the 2-metric dimension and 2-locating number can be made arbitrarily large. 2-locating sets in the join, corona, edge corona, and lexicographic product of two graphs have been characterized. From these characterizations, 2-locating numbers have been determined. This new invariant can also be studied for graphs under other binary operations. REFERENCES 1661 Acknowledgements The authors would like to thank the Department of Science and Technology - Acceler- ated Science and Technology Human Resource Development Program (DOST-ASTHRDP)- Philippines, MSU-Iligan Institute of Technology. References [1] R. Bailey, J. Cáceres, J. Garijo, A. González, A. Márquez, K. Meagher, and M.L. Puertas. Resolving sets for johnson and kneser graphs. European Journal of Combi- natorics, 34:736–751, 2013. [2] R. Bailey and I. Yero. Error-correcting codes from k-resolving sets. Discussiones Mathematicae, Graph Theory, 39:341–355, 2019. [3] J. A. Bondy and U. S. R. Murty. Graph Theory. Springer, 2008. [4] F. Buckley and F. Harary. Distance in Graphs. Addison-Wesley, Redwood City, CA, 1990. [5] J. Cabaro and H. Rara. On 2-Resolving Sets in the Join and Corona of Graphs. European Journal of Pure and Applied Mathematics, 14(3):773–782, 2021. [6] J. Cabaro and H. Rara. On 2-Resolving Dominating Sets in the Join and Corona and Lexicographic Product of Graphs. European Journal of Pure and Applied Mathemat- ics, 15(3):1201–1210, 2022. [7] S. Canoy Jr and G. Malacas. Locating Dominating Sets in Graphs. Applied Mathe- matical Sciences, 8(88):4381–4388, 2014. [8] S. Canoy Jr and G. Malacas. Locating Sets in a Graph. Applied Mathematical Sciences, 9:2957–2964, 2015. [9] A. Mahistrado and H. Rara. On 2-Resolving Hop Dominating Sets in the Join and Corona and Lexicographic Product of Graphs. European Journal of Pure and Applied Mathematics, 15(4):1982–1997, 2022. [10] A. Mahistrado and H. Rara. Outer-Connected 2-Resolving Hop Domination in Graphs. European Journal of Pure and Applied Mathematics, 16(2):1180–1195, 2023. [11] A. Mahistrado and H. Rara. Restrained 2-Resolving Hop Domination in Graphs. European Journal of Pure and Applied Mathematics, 16(1):286–303, 2023. [12] D. Managbanag and H. Rara. Forcing 2-Metric Dimension in the Join and Corona of Graphs. European Journal of Pure and Applied Mathematics, 16(2):1068–1083, 2023. [13] J. S. Mohamad and H. Rara. On Resolving Hop Domination in Graphs. European Journal of Pure and Applied Mathematics, 14(3):1015–1023, 2021. REFERENCES 1662 [14] J. S. Mohamad and H. Rara. 1-Movable Resolving Hop Domination in Graphs. Eu- ropean Journal of Pure and Applied Mathematics, 16(1):418–429, 2023. [15] J. S. Mohamad and H. Rara. Strong Resolving Hop Domination in Graphs. European Journal of Pure and Applied Mathematics, 16(1):131–143, 2023. [16] B. Omamalin, S. Canoy, and H. Rara. Locating Total Dominating Sets in the Join, Corona and Composition of Graphs. Applied Mathematical Scciences, 8(48):2363– 2374, 2014. [17] V. Saenpholphat and P. Zhang. On connected resolvability of graphs. Australian Journal of Combinatorics, 28:26–37, 2003. [18] S. Seo and P. Slater. Open Neighborhood Locating-Dominating Sets. Australian Journal of Combinatorics, 46:109–119, 2010. [19] P. Slater. Dominating and Reference Sets in a Graph. Journal of Mathematics and Physical Science, 22(4):445–455.